The relation between χ2 distributions and Gamma distributions, and Γ functions. Re- call that the Gamma distribution is one of the dis- tributions that comes up in the Poisson process, the others being the exponential distribution and the The sample variance and Poisson distribution. A Poisson process is when 2 events occur uniformly at random over time at a the χ distribution constant rate of λ events per unit time. The time T Math 218, Mathematical Statistics it takes for the rth event to occur has what is called D Joyce, Spring 2016 a Gamma distribution with parameters λ and r.

The sample variance for a sample X1,X2,...,Xn The gamma distribution also has applications is sometimes defined as when r is not an integer. For that generality the n factorial function is replaced by the gamma func- 1 X S2 = (X − X)2 tion, where n − 1 i i=1 Z ∞ x−1 −t but sometimes as Γ(x) = t e dt. 0 n 1 X S2 = (X − X)2. The gamma function and a related function called n i i=1 the beta function were invented by Euler in 1729. One of the main purposes of the gamma function is We’ll use the first, since that’s what our text uses. to generalize the factorial function to nonintegers. In the same way that the normal distribution is When r is a nonnegative integer, used in the approximation of means, a distribution 2 called the χ distribution is used in the approxima- r! = Γ(r + 1) tion of variances. Let Z1,Z2,...,Zν be ν independent standard so that 1 = 0! = Γ(1), 1 = 1! = Γ(2), 2 = 2! = Γ(3), normal variables. Then the sum of their squares 6 = 3! = Γ(4), etc. In fact, Γ(x) is defined except

2 2 2 when x = 0, −1, −2, −3,.... It’s also defined for X = Z1 + Z2 + ··· + Zν complex numbers. The gamma function has several has what is called a χ2 distribution with ν degrees properties including these of freedom, written χ2, or more simply χ2 when ν ν Γ(x + 1) = xΓ(x) is understood. The letter ν is the Greek letter nu, π and it is often used for the number of degrees of Γ(x)Γ(1 − x) = sin πx freedom. √ Γ(x)Γ(x + 1 ) = 21−2x π Γ(2x) The main purpose of a χ2 distribution is its rela- 2 √ Γ( 1 ) = π tion to the sample variance for a normal sample. 2 Suppose the sample X1,X2,...,Xn is from a nor- It turns out that the χ2 with ν degrees of freedom 2 mal distribution with mean µ and variance σ , then is gamma distribution with a fractional value for r, 2 2 the sample variance S is a scaled version of a χ namely r = ν/2, and λ = 1/2. That implies that distribution with n − 1 degrees of freedom 2 the density function for a χν distribution is (n − 1)S2 ∼ χ2 . xn/2−1ex/2 σ2 n−1 f(x) = , for x ∈ [0, ∞), 2n/2Γ(n/2) The details of the proof are given at the end of section 5.2 of the text. its mean is µ = n, and its variance is σ2 = 2n.

1 Our text has a table of values for the χ2 distri- bution, Table A.5 on page 676. We’ll work through example 5.5 on page 178.

2